Higher order Lipschitz Sandwich theorems

Journal article

We investigate the consequence of two Lip(γ) functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given K0 > ε > 0 and γ > η > 0 there is a constant δ = δ(γ, η, ε, K0) > 0 for which the following is true. Let Σ ⊂ R d be closed and f, h : Σ → R be Lip(γ) functions whose Lip(γ) norms are both bounded above by K0. Suppose B ⊂ Σ is closed and that f and h coincide throughout B. Then over the set of points in Σ whose distance to B is at most δ we have that the Lip(η) norm of the difference f − h is bounded above by ε. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip(γ) functions f and h are only close in a pointwise sense throughout the closed subset B. We require only that the subset Σ be closed; in particular, the case that Σ is finite is covered by our results. The restriction that η < γ is sharp in the sense that our result is false for η := γ

Authors: Lyons, T; McLeod, AD

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Journal of the London Mathematical Society
• Volume: 111
• Issue: 3
• Article number: e70121
• Publication date: 2025-03-07
• Acceptance date: 2025-02-19
• DOI: 10.1112/jlms.70121
• EISSN: 1469-7750
• ISSN: 0024-6107
• Language: English